(679f) Computing Interval Bounds On the Solutions of Nonlinear Index One DAEs

A method is presented for computing tight interval bounds on
the solutions of nonlinear semi-explicit, index one differential-algebraic
equations (DAEs) subject to given intervals of permissible initial conditions
and parameters. Differential-algebraic equations are used to model a wide variety
of important dynamic chemical processes through the combination of material and
energy balances, thermodynamic relationships, reaction kinetics and empirical
correlations. Moreover, such models often contain parameters which are not
known precisely, such as physical constants, factors in empirical correlations,
process disturbances and model uncertainties. Accordingly, one often requires
information about an entire family of possible solutions, parameterized by
these uncertainties, rather than a single nominal solution.

Assuming that the uncertain or unknown parameters of
interest lie within known intervals, the proposed method computes time-varying
upper and lower bounds on the solutions of the given DAEs attainable with
parameters in these intervals. Similar methods for bounding the solutions of
ordinary differential equations (ODEs) have been studied extensively, with
applications in uncertainty analysis, state and parameter estimation, safety
verification, fault detection, global optimization, validated numerical
integration, and controller synthesis. On the other hand, producing guaranteed
bounds on the solutions of DAEs has received much less success and remains an
open challenge.

The proposed approach for bounding DAE solutions combines
concepts from differential inequalities and interval Newton-type methods. The
first key result is an interval inclusion test which verifies the existence and
uniqueness of DAE solutions over a given time step, and provides a crude
interval enclosure. This test combines a well-known interval inclusion test for
solutions of ODEs (used in standard Taylor methods) with an interval inclusion
test for solutions of systems of nonlinear algebraic equations from the
literature on interval Newton methods. The second key result is a set of
sufficient conditions, in terms of differential inequalities, for two
time-varying trajectories to bound the differential state variables; i.e.,
those state variables whose time derivatives are given explicitly by the DAE
equations. Using this result, refined, time varying bounds on the state
variables over the given time step are computed using a technique which
simultaneously applies differential inequalities to the differential variables
and an interval Newton-type method to the algebraic variables.

Based on these key results, an efficient numerical
implementation using interval computations and a standard numerical integration
code is presented. Strengths and shortcomings of the proposed algorithm are
discussed in the context of a DAE model for a simple distillation process.